Naive diversification

As we previously discussed – in the part on Markowitz portfolio selection – investors are able to construct portfolios with better return-risk profiles by combining a large number of securities. This generally leads to a better performance than a single or small collection of securities can offer. However, the Markowitz model is complicated, sensitive to small changes in the parameters, and often infeasible in practice. A naive diversification approach on the other hand, is very simple and easy to implement. At the same time, it still honors the key message of Modern Portfolio Theory (MPT) that is diversification.

Naive diversification – the theory

The following equation shows a ‘1-factor market model’ that describes the excess return of a security as a function of its ‘abnormal’ return, α, its systematic risk β towards the market, and its idiosyncratic risk ε

 r_i - r_f = \alpha_i +  \beta_i \cdot (r_m - r_f) + \epsilon_i

When we construct a portfolio from multiple assets, our portfolio return can be described as

 r_p - r_f = \alpha_p +  \beta_p \cdot (r_m - r_f) + \epsilon_p

Suppose we increase the number of securities a lot. In that case the above equation reduces to:

 r_p = r_f +  \beta_p \cdot (r_m - r_f)

This is because both the ‘abnormal’ return α and firm-specific risk ε disappear.At the same time, systematic risk component (β) converges to 1 (i.e. equals market risk).

 \alpha_p = \lim_{i \to \infty}\dfrac{1}{N}\sum_{i=1}^{N}\alpha_i = 0

 \beta_p = \lim_{i \to \infty}\dfrac{1}{N}\sum_{i=1}^{N}\beta_i = \beta_m = 1

 \epsilon_p = \lim_{i \to \infty}\dfrac{1}{N}\sum_{i=1}^{N}\epsilon_i = 0

In other words, investors that wish to construct a proper portfolio don’t need to involve themselves with a complicated approach such as MPT. Instead, they are able to form an ‘optimal’ by just investing (equally) in a large number amount of assets. The portfolio is called a 1/n portfolio. Such a portfolio will, theoretically, have an exposure to market risk that is equal to 1. Investors who are risk-averse and that are content with lower levels of expected return, can decide to invest only part of their portfolio in the 1/n portfolio, and keep the rest in cash.

As long as the amount of securities is large enough and investment in any single security is not overstated, investors can obtain, through naive diversification, the market return.  Although an equal-weighted portfolio is the most common way to apply naive diversification, the theory actually is independent of the amount invested in any single security. At least, as long as the number of securities is large enough. In practical terms, this underlying idea initially contributed to the emergence of ETFs in the 90s.

Advantages of naive diversification

Naive diversification has several advantages. First, unlike a lot of other approaches available to investors, constructing a naive portfolio is not computationally intensive. This is because there are no parameters that have to be estimated. Second, and related to the first advantage, the approach is not sensitive to estimation error. As such, a naive portfolio is a more robust approach to investing. Finally, research suggests that a naive portfolio in practice does almost equally well as more sophisticated approaches.

Summary

Investing in a large number of securities allows investors to reduce overall portfolio risk. As long as the number of investments is sufficiently high, the return on any single security will not affect the risk-return profile of the entire portfolio. Naive diversification, by investing using arbitrarily chosen weights, (e.g. equally-weighted, value-weighted,…), will thus perform sufficiently well if enough assets are included in the portfolio.  This portfolio construction can easily be done using an Excel spreadsheet.

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